In my talk I will consider the short rate equation of the form
\begin{gather} \label{eqqq}
\dd R(t)=F(R(t))\dd t+\sum_{i=1}^{d}G_i(R(t-))\dd Z_i(t), \quad R(0)=R_0\geq 0,\quad t>0,
\end{gather}
with deterministic functions $F,G_1,...,G_d$ and a multivariate L\'evy process $Z=(Z_1,...,Z_d)$ with possibly dependent coordinates.
It is supposed to have a nonnegative solution which generates an affine term structure model. Under some mild assumptions on the L\'evy measure of $Z$ it appears that the same term structure is generated by an equation with affine drift term and noise being just a one-dimensional $\alpha$-stable process with index of stability $\alpha\in(1,2)$\DSEMIC this generalizes the classical results on the Cox-Ingersoll-Ross (CIR model), as well as results on its extended version where $Z$ is a one-dimensional L\'evy process.

Further, for such a model I will characterize the possible shapes of simple forward curves. A description of normal, inverse and humped profiles in terms of the equation coefficients and the stability index $\alpha$ will be provided.


I will also show that there exist other affine term structure models in which the short rate satisfies \eqref{eqqq}, and fully characterize them under the assumption that the coordinates of $Z$ are independent and have regularly varying Laplace transforms.

The talk is based on results obtained together with Micha{\l} Barski (University of Warsaw), see arXiv:2407.21425 and arXiv:2402.07503. Some of these results are to appear in Modern Stochastics: Theory and Applications.